Alright, so I am not a doctor, but from what I know and have experienced with allergies: Yep, definitely possible. Especially if there is sanding and resin dust involved if you are asthmatic (not quite an allergy, but definitely an adverse reaction). Resin is a type of plastic, which you can be allergic to. Basically, in theory you could be allergic to anything with a protein of some sort. I don't know much about resin or plastics as a whole (because getting into the really detailed parts kind of seems to require a healthy understanding of chemistry, which I do not have) but I think there are a lot of different mixes and grades and types of resin that it could be difficult to be sure whether you were allergic or not to all of them. Some might be better or worse, and everyone would react somehow if it was a toxic type of resin like recasts might be made of. This is something that I wonder about a lot, as I have a lot of allergies, some of them to certain plastic poly- whatevers, which resin falls into. Especially since once when I was young I was in a shop with all sorts of older collectable resin dolls in it...and I literally could not breathe, at all. I think there might have been some resin dust (or a lot of resin dust) in the air and the dolls might have been old enough to maybe not have had completely non-toxic resin, but because of that I am still super paranoid about buying a resin doll and then discovering I can't breathe around the doll. I think it may just be some sort of resin I don't do well with, since I have one art doll which has a faux fur body and a resin face which, despite my extreme worries when it came in the mail, I am (thankfully!!) completely fine with that particular resin, and I handle that doll loads.
So, it's a tricky subject and I am definitely not qualified or knowledgeable enough to answer all of your questions without doubt, but to answer your titular question: yes, it is possible to be allergic to resin. There are loads of other messy variables which I don't know about though. Also, keep in mind that just because it is possible...doesn't mean it will happen. Thankfully, or else I would have been killed by a chair leg impaling me a long time ago.
EDIT: I'm allergic to polyester (so can't wear any clothes with it or I get a very unpleasant rash) just to let you know that, though unusual, there are people who are allergic to different poly plastics. I just have a very ...
special auto-immune system. -.- *cough*crazyparanoid*cough*
Anyways--dolly cuddling time!